Mini Course: Okinawa Lectures on Entropy

Please REGISTER here if you plan on coming...

Fridays 1000-1200 from April 10 to June 12 in Lab 5 D23

Audience: Theoretical physicists and mathematicians interested in entropy

Prerequisites: Undergraduate probability theory, functional analysis, and quantum theory 

Description: After a historical and conceptual introduction, the most important classical and quantum entropies and relative entropies will be introduced. My approach to classical entropies will be based on the theory of large deviations in probability theory, whereas quantum entropies will be studied in the context of von Neumann algebras. These topics will be introduced from scratch, but in a way that should also be interesting to those already familiar with them. Although my interest in entropy originated in black hole thermodynamics, and the material should be highly relevant to this, the main application of entropy in this course will be to hypothesis testing, which beautifully links the classical to the quantum theory and is important for example in quantum information theory. The quantum phenomenon of entanglement will also be studied in this context. The course tries to be both mathematically rigorous and interesting to theoretical physicists. For example, proofs of theorems may be replaced by examples that show what is going on. I hope to write (preliminary) lecture notes on the go, available soon after each lecture.

Preliminary outline of the lectures:
1. Introductory overview: history and outline of entropy, large deviations, von Neumann algebras, and their connections (slides)
2. Classical entropies from the point of view of asymptotics and large deviations
3, Large deviation theory: theorems of Sanov, Cramér, Gärtner-Ellis, and Varadhan
4. Classical hypothesis testing: Neyman-Pearson lemma and theorems of Chernoff, Stein, and Hoeffding
5. Von Neumann entropy and quantum relative entropy in finite-dimensional Hilbert spaces
6. Quantum hypothesis testing in finite-dimensional Hilbert spaces
7. Basic von Neumann algebra theory, connection with probability and entanglement
8. Tomita-Takesaki (modular) theory and associated quantum entropies
9. Crossed products and noncommutative L^p-spaces over von Neumann algebras
10. Quantum hypothesis testing in general von Neumann algebra